Magnetic propulsion of microspheres at liquid-glass interfaces
Geir Helgesen
Citation: Journal of Applied Physics 123, 064902 (2018); doi: 10.1063/1.5011350 View online: https://doi.org/10.1063/1.5011350
View Table of Contents: http://aip.scitation.org/toc/jap/123/6 Published by the American Institute of Physics
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Magnetic propulsion of microspheres at liquid-glass interfaces
GeirHelgesena)
Physics Department, Institute for Energy Technology, NO-2027 Kjeller, Norway
(Received 31 October 2017; accepted 25 January 2018; published online 13 February 2018) Bio-coated, magnetic microspheres have many applications in biotechnology and medical technology as a tool to separate and extract cells or molecules in a water solution by applying external strong magnetic field gradients. However, magnetic microspheres with or without attached cargo can also be separated in the liquid solution if they are exposed to alternating or rotating, relatively weak magnetic fields. Microspheres that have a higher density than the liquid will approach the bottom surface of the sample cell, and then a combination of viscous and surface frictional forces can propel the magnetic microspheres along the surface in a direction perpendicular to the axis of field rotation. Experiments demonstrating this type of magnetic propulsion are shown, and the forces active in the process are discussed. The motion of particles inside sample cells that were tilted relative to the horizontal direction was studied, and the variation of propulsion velocity as a function of tilt angle was used to find the values of different viscous and mechanical parameters of motion. Propulsion speeds of up to 5lm/s were observed and were found to be caused by a partly rolling and partly slipping motion of rotating microspheres with a slipping coefficient near 0.6.VC 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/
licenses/by/4.0/).https://doi.org/10.1063/1.5011350
I. INTRODUCTION
In magnetophoresis, paramagnetic or ferromagnetic micro- and nanoparticles that are dispersed in a fluid are forced to move due to external magnetic field gradients.1 Such magnetic microbeads are often made by emulsion polymerization, and also, a wide range of other preparation techniques exist.2The magnetic propulsion force acting on microbeads will depend on their magnetic momentm, which is normally proportional to their volume, the strength of the externally applied fieldH, and the field gradientrH, and it will pull them into regions of stronger field. For small par- ticles, this process is very slow due to the competition between magnetic propulsion, viscous forces, and Brownian motion. However, by applying relatively weak, rotating mag- netic fields and utilizing microsphere-surface interaction forces, the magnetic propulsion speeds can be controlled and possibly increased relative to applying static magnetic fields.
This method will be discussed below after a summary of the current status of magnetic propulsion.
In some applications, micro-swimmers can propel due to their ability to convert chemical energy or heat in their surroundings into a directed motion.3 However, for micro- meter sized particles, the speeds will be low due to viscous effects since the Reynolds numberReis very small (Re1).
For applications in microfluidics and biotechnology, the liquid volumes are usually also small, which means that the magnetic particles are close to walls. The effect of a wall is normally to slow down any motion due to an increased effective fluid viscosity near the wall. In some cases, the presence of a wall can also be utilized in order to enhance
the propulsion. The theory of motion of particles near solid walls has been discussed in several papers.4–6By applying rotating magnetic fields, magnetic microbeads can be forced to rotate.7–10 The motion of active and passive rotors has been discussed by Fily et al.,11 and magnetic colloidal sur- face walkers12 and tumblers13 have been reported.
Biocompatible ferrofluids inside microfluidic platforms have been used to manipulate and separate out nonmagnetic microparticles and live cells.14The current status of how to activate soft matter with magnetic torque has been reviewed by Erbet al.15
The motion of small suspended particles near walls can involve competition among viscous forces, frictional forces, electrostatic forces, and gravitational forces. The rolling motion of non-colloidal spheres down inclined planes inside fluids has been discussed in a couple of papers. Smartet al.16 made a theoretical model and did experiments for rough spheres rolling on planes at various inclinations and showed the effect of surface roughness. Galvinet al.17did a similar study using spheres that had two separate scales with distinct distributions of the surface roughness elements. They man- aged to find a transition from smaller to larger separation heights above the inclined plane, and this was dependent on the inclination angle and the roughness scales. The inertial lift force on a rigid sphere in a linear shear flow field near a wall was calculated by Cherukat and McLaughlin.18 Later, Krishnan and Leighton19and King and Leighton20extended these calculations and measured the transition from rolling to translation with slip along the surface. The effects of inter- action between a microparticle in shear flow and charged surface patches have also been studied.21 Agayan et al.22 used optical tweezers to trap magnetic microspheres near a surface and exposed the particles to rotating magnetic fields.
0021-8979/2018/123(6)/064902/8 123, 064902-1 VCAuthor(s) 2018.
JOURNAL OF APPLIED PHYSICS123, 064902 (2018)
The particles then got a new equilibrium position, which was displaced relative to the center of the optical trap, and the authors studied how the rolling and slipping of the micro- spheres depended on the magnetic rotation rate and surface properties. They also measured the same for free, non- trapped microspheres on a glass surface and observed a shift in the direction of motion of the particles relative to the plane of the rotating field in both cases. This was interpreted as a shift of the direction of the spheres’ magnetic moment away from the plane of rotation. Recently, it was proposed that the motion of magnetic beads on inclined planes can be used as a technique for size separation of the beads.23 Martinez- Pedrero and Tierno8 showed that carpets of magnetically driven rotating beads can be used to carry cargo such as cells and modeled the translational motion as purely due to hydro- dynamic effects.9In the present study, it will be shown that surface contact forces also play an important role in near- surface propulsion.
II. EXPERIMENTAL DETAILS
The experimental setup consisted of an optical micro- scope (Nikon Optiphot), a custom made set of three orthogo- nal pairs of current carrying coils, and a custom made computer controlled power supply for the magnet coils. A digital-to-analogue converter (Measurement Computing Systems), which was controlled by a PC, supplied the input signals for the power supply. The oscillating digital signals were generated in Labview. The microparticle motion was recorded on a PC using a C-mounted DinoEye USB video camera and the DinoCapture software. The amplitudes and phases of the coil currents were monitored using a digital oscilloscope (GW-INSTEK) and three separate multimeters.
Images from the videos were extracted at specific times using the Cyberlink PowerDirector program. Particle tracks were also extracted directly from AVI-video files using the Video-spot-tracker-v08 software (CISMM at UNC Chapel Hill,www.cs.unc.edu/~nanowork/cismm/). Figures 1(a)–1(c) show the coil system, the glass sample cell, and the micro- spheres inside.
The magnetic microparticles were dispersed in de-ionized water containing 0.1% SDS (Sodium Dodecyl Sulfate) surfac- tant and placed in between a microscope glass slide and a cover slide. The separation between the slides was controlled by using one or two layers of double-sided tape as spacers along two of the sides of the cell (giving a plate separation of
h85lm or 170lm). All the sides of this thin cell were sealed using epoxy glue. Typical coil current amplitude used was I¼1.4 A that gave a magnetic field amplitude of about H¼1100 A/m.
The microspheres used in the current experiments were provided by the Ugelstad laboratory at NTNU/SINTEF (Trondheim, Norway). They were made of polystyrene, had an iron content of about 24% (magnetite Fe3O4/maghemite Fe2O3), and had a density of qs1.6 g/cm3. Spheres of diametersd¼30lm, 4.7lm, 3.5lm, and 1.5lm were used.
Similar particles have been characterized in more detail by Fonnumet al.,24who found paramagnetic mass susceptibili- ties in the range of vm¼55–100105 m3/kg, and these authors also found that the microspheres contained magnetic nanoparticles of size about 8 nm with inter-particle magnetic interactions, thus giving rise to nanoparticle clusters.
Sample cells containing a very dilute mixture of micro- spheres were mounted in the coil system, and the motions of particles with and without a rotating magnetic field were recorded for several inclinations of the whole microscope system. A simple tilt table was used for this purpose. The recorded videos were used to extract propulsion velocities.
All velocity measurements were repeated several times using different microspheres within a sample and also using differ- ent sample cells. The average velocity was found to vary slightly among equally sized particles. Thus, the reported data correspond to average velocity values with standard deviations for individual microspheres.
III. THEORETICAL CONSIDERATIONS
Janssenet al.7have shown that the average angular rota- tion frequency of a microbead,xb, depended linearly on the magnetic field rotation frequencyfH¼xH/2pup to a critical frequency fHcrit. Above that, xb decreased gradually with increasing fH. This result is similar to the frequency depen- dence previously found for pairs of so-called magnetic holes.25,26 Janssen et al. also showed that the permanent magnetic momentmpseems to be locked to one direction of a fixed axis within the sphere. For higher frequencies, they observed that the main reason for sphere rotation was the induced magnetic moment caused by the imaginary part of the complex susceptibilityv00ðxHÞ. The angular directionub of a fixed axis within the bead could be modeled as
mpl0HsinðxHtubÞ þv00ðxHÞl0H2V¼kdub dt ; (1)
FIG. 1. (a) The coil system used in the present study for creating three separate magnetic fields that can rotate in theXY-,XZ-, orYZ-planes. Part of the sample cell can be seen inside the coils. (b) The sealed sample with microspheres on a 25 mm wide glass slide. (c) Microscopy image of the 4.7lm magnetic micro- spheres. (d) AFM surface scan of a dry microsphere. Similar scans were used to find the surface roughness (see Part IV). This scan has been corrected for the spherical shape of the particle.
064902-2 Geir Helgesen J. Appl. Phys.123, 064902 (2018)
withVbeing the volume of the bead and ka hydrodynamic drag factor. For magnetic monodisperse nanoparticles, the imaginary part of the susceptibility is
v00ðxHÞ ¼v0 xHsm
1þx2Hs2m; (2) withv0being the static field susceptibility andsmthe mag- netic relaxation time, which is mainly controlled by the nano-domain anisotropy energy. Inside the beads, there was a broad distribution of nanoparticles, which was assumed to be log-normal by the authors of Ref.7, and this was inter- preted as the reason for a broad peak in bead average rotation frequencyX¼ hdudtbiat higher field rotation frequencies.
For microparticles in the present size range, the most relevant parameter for characterizing their motion is the Reynolds number of sedimentation, i.e., for a sphere of radiusafalling freely in a liquid far away from disturbing walls,Res¼Usa. Here,Us¼29Dqga2=g0is the sedimentation velocity of the sphere withDq¼qs –ql¼600 kg/m3being the density difference between microspheres and water in the present case, g¼9.81 m/s2the acceleration of gravity, and g0¼0.80 mPa s and¼8.0107m2/s the dynamic viscos- ity and kinematic viscosity, respectively, of water at tempera- tureT¼30C. From this,Us¼9.0lm/s andRes¼3105. Thus, all motions are strongly damped.
When a non-rotating sphere is translated at a velocityUx in thex-direction parallel to a wall, there will be a viscous drag force acting in the direction opposite to the motion27
Ftx¼ 6pag0fxxt Ux; (3) with a translational drag correction coefficient fxxt ¼fxxt sa that is dependent on the distancesfrom the sphere center to the wall. Similarly, for a sphere that is kept fixed, but is rotating at angular velocity Xy about the y–axis direction, there will be a viscous force27
Frx¼6pa2g0fxyr Xy (4) that is trying to translate the sphere along the x-direction.
Here,fxyr ¼fxyr asis the corresponding rotational-translational drag correction coefficient.
For a sphere rotating very close to the wall with a separa- tion ofd¼s–abetween the surface of the sphere and the flat surface of the wall, the correction factor due to translation, fxxt, and the correction factor due to rotation, fxyr, can be approximated by28,29
fxxt 8 15ln a
d þ0:9588¼ 8 15ln s
a1
þ0:9588; (5)
fxyr 2 15ln a
d 0:2526¼ 2 15ln s
a1
0:2526: (6) As an example, the factors fxxt and fxyr increase from 2.56 and 0.147, respectively, at da¼0:05 to 5.87 and 0.975 at
d
a¼104.
Now consider the situation with a magnetic microsphere located near a planar surface, which is tilted an angle a
relative to the horizontal direction and with thex–axis of the coordinate system pointing along the surface in the tilt direc- tion, denoted “uphill” from here. This is shown in Fig. 2.
They–axis is along the surface in the perpendicular in-plane direction. The sphere is acted on by an external, rotating magnetic field that is forcing it to move up or down the inclined plane. The forces acting on the sphere (see Fig. 2) are gravityWg ¼43pa3Dqg, a surface normal forceNcin the contact point between the sphere and the surface, a lift force Lhdue to a hydrodynamic lift at the finite Reynolds number, a friction forceFf¼lkNcacting in the contact point parallel to the surface with lkbeing the kinetic friction coefficient, and a viscous drag force Fhdue to the translation and rota- tion of the sphere. This drag is opposite to the direction of motion. The equations for balance of forces parallel and perpendicular to the surface areeFfþFhWgsina¼0 and NcþLhWgcosa¼0 with e¼ þ1 for a sphere moving uphill ande¼–1 for a sphere moving in the opposite direc- tion, i.e., downhill. Combining these equations, one finds
elkðWgcosaLhÞ þFhWgsina¼0: (7) The drag force is due to translation and rotation of the sphere Fh¼FthþFrhwith the latter two given by Eqs.(3)and(4).
Now assume that the sphere is rotating with a positive angular frequency Xy, having an uphill velocity Ux. The motion is partly rolling and partly slipping, and this
“skipping” can be modeled through a slipping coefficient c asUx¼caXy.23Then
Fh¼6pg0a fxxt Uxþfxyr aXy
¼ 6pg0aUx fxxt 1 cfxyr
: (8)
When a rigid sphere is moving near a flat surface, there will be a small lift force on the sphere pushing it away from the surface in the surface-normal z–direction. This will happen when the surface is within the disturbance flow-field of the sphere.18In general, if a particle is rotating and translating near a wall, the hydrodynamic lift forceLhwill be
LhðX;cÞ ¼qa4X2 k2k3cþk1c2
; (9)
where k1¼1.755, k2¼0.546, and k3¼–2.038 are the lift coefficients.19,23 Combining Eqs. (7) and(8), one finds the force balance23
FIG. 2. Schematic diagram of a microsphere moving at velocityUxnear an interface that is tilted at an anglea. The forces acting are gravitational force Wg, surface normal contact forceNc, surface friction forceFf, hydrodynamic lift forceLh, and viscous drag forceFh.
elkðWgcosaLhÞ 6pg0aUx fxxt 1 cfxyr
Wgsina¼0:
(10) Inserting values for tilt angle a(5a30), slipping coefficient 0c1, and angular velocity of rotation X 1 s1, one finds that the lift forceLhis typically a factor of 107 smaller than the gravitational force Wg0.3 pN and thus can be neglected. Then, the velocity of the microspheres is given by
Ux¼Uselkcosasina fxxt 1
cfxyr
; (11)
where Us is the sedimentation velocity defined above and with e¼ þ1 for uphill motion and e¼–1 for downhill motion. For small tilt anglesa, one finds a linear Ux vs. a relationshipUxft Us
xx1cfxyr ðelkaÞ.
As will be shown below, the uphill motion vanishes, Ux¼0, at a certain critical angleacrit, and then c¼0. Then, the force balance Eq.(10)is replaced byWgðlkcosasinaÞ 6pg0a2fxyr Xy¼0, and the kinetic friction coefficient can be found as
lk¼tanacritþ a Us cosacrit
fxyr Xy: (12)
IV. RESULTS AND DISCUSSION
Figure 3 shows recorded traces for the motion of two microspheres moving concomitantly inside a horizontal glass
sample cell of thickness about 85lm (data extracted from Video 1 in the supplementary material). The direction of motion is dominantly parallel to the plane formed by the rotating magnetic field, but there is also a small fluctuating component perpendicular to this plane. This may partly be due to Brownian motions and partly due to tiny obstacles on the glass surface. For much smaller particles (d1.5lm), the Brownian component was dominating and the propulsion was very weak for the magnetic field strengths used in the current study (H<2 kA/m). Some variations in the velocities among equally sized particles were observed, which may be attributed to variations of the amount of paramagnetic mate- rial (magnetite/maghemite) inside each sphere. Also, perma- nent sticking of a few spheres to the glass surface was observed. The ability of bringing particles almost back to their starting position after being moved around for 200 s (825 s), as shown in Fig.3, indicates the level of precision for control of the particles using the current method.
It may be noted that after exposure to the magnetic field, some clustering of microspheres could be seen, and this leads to collective propulsion of rafts of particles as has recently been reported8or to conveyor belt modes.9In order to avoid collective modes or hydrodynamic coupling of several spheres, only microspheres that were well separated from each other were studied.
First, the magnetic field strength and field rotation fre- quency dependencies of microsphere propulsion velocity were explored. Figure 4 shows the results for d¼4.7lm spheres inside horizontal sample cells. The propulsion veloc- ity increases up to aboutfH¼10 Hz and was nearly constant above that. Even at the lowest frequency of 1 Hz, the micro- sphere rotation was not following the field rotation, and the sphere was not rolling on the surface without slipping, since this would correspond to a propulsion velocity ofU¼pdfH
¼14.8lm/s. Thus, either the rotation of the microsphere did not follow in phase with the field rotation or the sphere was strongly slipping. Most likely, it was a combination of both, as will be demonstrated in the following. As seen in Fig. 4(a), there are some variations in the average velocity among particles of the same size, which as noted above may be due to variation in their content of the magnetic material.
As the frequency increased, the rotation of the spheres was clearly not able to follow the field, and the microspheres were acted on by a time-averaged torque from the magnetic field, which seems to be nearly frequency independent for fH>20 Hz. A field rotation frequency of fH¼100 Hz was used in the rest of this study. The observed frequency depen- dence is consistent with what have been reported by Agayan et al.,22 who found an increase in velocity for magnetic spheres of diameter 9lm for field frequency up to fH
¼2–3 Hz and beyond that a small decrease in velocity.
Figure 4(b) shows how the velocity depends on the field strengthH. According to Eq.(1),Ux/H2behavior might be expected for superparamagnetic microbeads with mp¼0.
Regression fit to the measured Ux vs. H data showed a slightly weaker dependence, Ux/H1.9. For stronger fields, H2 kA/m, Ohmic heating in the coil system prevented measurements. Janssenet al.7measured thatX/H2for the
FIG. 3. The traces of twod¼3.5lm spheres being propelled by a rotating magnetic field at the interface between water and a horizontal glass surface.
The field amplitude wasH¼1.1 kA/m rotating in theXZ- orYZ- planes, and the rotation frequencyfH¼100 Hz. The propulsion started by applying a rotating field in theXZ-plane of the coil system. After 25 s, the plane of rota- tion was changed to theYZ-plane for the next 25 s. This procedure was repeated using the opposite rotation direction within theXZ- andYZ-planes, thus bringing the particles in a continuous motion back to their starting posi- tions. This square loop motion was then repeated a second time. Note a small misalignment between theX- andY-axes of the coil system and those of the microscope camera. The inset images show the two particles (enlarged by 50%) near their starting positions. The particle velocity was found to be 2.660.2lm/s.
064902-4 Geir Helgesen J. Appl. Phys.123, 064902 (2018)
bead angular rotation at high frequency in their study of trapped magnetic beads.
The magnetic and viscous forces acting on particles in such micron-scale systems are difficult to measure directly.
However, this can be done indirectly by comparing them to a known reference force, which can easily be found or calcu- lated, such as the gravitational force on the particles. The importance of the effect of magnetic phase slip within the spheres7and the effect of slipping motion on the surface can be clarified when these forces are found. Calibrating the forces against gravity was done by placing the microscope with the coil system and sample cell on an inclined tilt table.
By tilting one direction of the sample cell (here denoted the X–direction) relative to the horizontal, the microspheres will experience a component of the effective gravitational force along the surface parallel to the direction of forced motion.
The propulsion velocities can then be compared to the effec- tive sedimentation velocity due to the gravitational field alone at the same tilt angle.
Figure 5 shows the X–position of microspheres as a function of time for samples at various tilt angles in the range ofa¼0–22.5, as was shown in the schematic setup in Fig.2. The position was measured along the tilt direction and relative to the lower edge of the microscope field- of-view. Here, the direction ofH–field rotation was such that the spheres were first propelled downhill, and after a time t¼10 s–30 s, the direction of rotation was changed to propel- ling uphill. This procedure was repeated until the particles left the lower edge of the field-of-view in the microscope. In some cases, the magnetic field was turned off and the spheres were allowed to move downhill by gravity force only. This is marked by the labels “H¼0” in the figure. An example of a video recorded during one such experiment can be seen in Video 2 in thesupplementary material. As can be seen, for small tilt angles, the spheres move downhill with a slightly higher speed than they move uphill, and this reduction of uphill velocity continues with increasing a until about a¼22.5when the uphill motion vanishes completely. Then,
the sphere was rotating in the same direction as the field (uphill,Xy>0) but the slipping was complete with slipping coefficient c0. The sphere was spinning “freely” or slightly sliding down due to gravity. At the critical angle acrit, the translational viscous drag force vanishes (Ux¼0), and only the uphill directed rotational-translational force Eq.
(4) remains, which together with the force of slipping fric- tion balances the in-plane component of gravity.
The results of these sample cell tilt experiments are pre- sented in Fig.6, which shows the propulsion velocityUxas a function of the cell tilt angle a. Velocities were calculated for uphill propulsion (black circles), downhill propulsion
FIG. 4. (a) Propulsion velocityUxvs. magnetic field rotation frequencyfH(logarithmic scale) for microspheres of diameterd¼4.7lm using a magnetic field strength ofH¼1.1 kA/m. Each data point and its error bar represent the average value and standard deviation based on several separate measurements for one single sphere. For each frequency, typically, the velocity of two to four microspheres was measured, and the scattering of velocity data at a frequency repre- sents the variation among similar spheres. (b) VelocityUxas a function of field strengthHatfH¼100 Hz for the same spheres shown in a double-logarithmic plot. The solid regression line shows aUx/H1.9behavior.
FIG. 5. Examples of the recorded traces (X-position vs. time) ford¼4.7lm spheres moving in a sample cell that was inclined at various anglesarelative to the horizontal direction. The spheres were propelled by a rotating magnetic field of frequencyf¼100 Hz and amplitudeH¼675 A/m. Here, the spheres first are propelled downhill (negativeX-direction,xH<0), and after a timet, the direction of field rotation was changed (xH>0) for the next time interval t. The values oftfor these curves weret¼60 s, 30 s, 30 s, and 10 s (top to bottom). The less steep parts at the beginning or end of two of the curves marked asH¼0 correspond to no driving force, i.e., only gravity acting.
(red triangles), and also for spheres drifting freely downhill due to gravity only (green squares). As can be seen, there is some scattering of the data around three nearly parallel lines, with only minor deviations for the pure gravity case and larg- est deviations for the downhill motion data. The solid lines are linear regression fits to the three datasets in the angular range of 0a<20, which may be considered the “small”
tilt angle range. For tilt anglea¼22.5, the data fall clearly below these small tilt angle regression lines, and this is most clearly seen in the case of downhill rotation (Xy<0).
As expected from the linearization of Eq.(11)for small a, the data fall on three separate lines, which using linear regression can be approximated by
UuðaÞ ¼½1:430:059aðdeg:Þlm
s ¼ð1:433:4aÞlm s ;
(13) UdðaÞ ¼ 1:43½ 0:048aðdeg:Þlm
s ¼ 1:43ð 2:8aÞlm s ; (14) UgðaÞ ¼ 0:049aðdeg:Þ½ lm
s ¼ 2:8alm
s ; (15) for uphill (u), downhill (d), and pure gravitational (g) motion, respectively.
Now, first consider the case when only gravity is acting (middle curve of Fig.6). Since the microspheres are in con- tact with the surface, it seems reasonable to assume that the microspheres are rolling downhill on the surface. Then,c¼1 (pure rolling), lk¼0, and e¼–1 (i.e., the component of gravity is in the direction of motion), and Eq. (11) gives Ux¼ Usftsina
xxfxyr Usfta
xxfxyr for small a. From the value ofUgin Eq.(15), one findsftUs
xxfxyr ¼2:8lms. Using the value of sedimentation velocity calculated above, Us¼9.0lm/s, thenfxxt fxyr ¼3:2, and from Eqs.(5)and(6), one finds that
the effective separation between the surface of the spheres and the glass surface wasd16 nm.
In order to check how this effective surface separation compares to the surface roughness of the microspheres, the surfaces of dry spheres deposited on a microscope slide were imaged using an atomic force microscope (AFM). A section of a typical AFM scanning image is shown in Fig. 1(d). As can be seen, the surface is not smooth on the nm-scale but looks bumpy with many protuberances. The scans showed local deviations from the spherical shape typically in the range of 5–30 nm. Based on such AFM scans, the surface roughness (locale deviation from spherical shape) can be estimated to about 20 nm (RMS-value). In a similar way, the surface of a microscope glass slide was checked and was much smoother with all protuberances well separated and typically less than 5 nm in height. These roughness results are in good agreement with the effective surface separation value found from the hydrodynamic interactions. Thus, the effective hydrodynamic separation of the surfaces in these experiments wasd15–20 nm. Due to the larger protuberan- ces, the spheres and the glass surface were in direct contact, and the frictional force was the main cause for propulsion when the magnetic torque was rotating the spheres.
For small tilt angles a, Eq. (11) can be linearized as UxðaÞ ¼AþBawithA¼ftelkUs
xx1cfxyr andB¼ ft Us
xx1cfxyr. Due to the scattering of the individual data points in Fig. 6, the slopes of the regression lines for uphill and downhill propul- sion are equal within the statistical uncertainty, and their average value was used in the subsequent analysis. Based on the effective surface separation dfound above, the drag cor- rection coefficients fxxtðdÞ and fxyrðdÞ were calculated. From this, the slipping coefficient c for driven uphill or downhill motion was found to bec¼0.59, and the angular rotation fre- quency of the microspheres was XyðfH¼100 HzÞ ¼Uxða¼ 0Þ=ca¼1:1 s1. Using the expressions forAandB, the fric- tion coefficientlkwas found to belk¼ jABj ¼0:49.
The regression lines in Fig.6are not good approxima- tions to the experimental results for values of a>20. However, due to the relative large scattering of data values, using the full expression Eq.(11)in the fitting and the whole range of a-values did not significantly change these results.
As can be seen in Fig. 6, the uphill propulsion vanishes, Uu¼0, at about a21, and for larger tilt angles, even microspheres with positive Xy are slowly sliding downhill.
Choosinga¼acrit¼21and using Eq.(12), the kinetic fric- tion coefficient can be estimated to belk0.51, which is in quite good agreement with the value lk¼0.49 found from the A/B-ratio. It may be noted that the exact values of slipping coefficientcand angular velocityXydepend on the effective separation d between the microspheres and the glass surface. A 25% increase in the value ofdgives 11%
increase incand similar reduction inXy.
For a fixed tilt angle, the main reasons for different average velocities of similar spheres are as follows: (i) the distribution of microsphere diameters d, which is about
Dd
d <1:5%;24 (ii) a variation in their induced magnetic moments mdue to size variation and to the distribution of the magnetic material inside; and (iii) differences in the
FIG. 6. The velocity ofd¼4.7lm spheres being propelled uphill (black circles) or downhill (red triangles) by aH¼675 A/m field rotating at a fre- quency of 100 Hz for different values of the sample tilt anglea. The green squares show velocity of downhill sliding motion due to gravity. Solid lines are linear regression fits. Each data point represents several measurements using the same microsphere. For some tilt angles, two or three different par- ticles were measured.
064902-6 Geir Helgesen J. Appl. Phys.123, 064902 (2018)
effective sphere surface to glass surface separation caused by the surface roughness. Variation of velocity due to the glass surface could also easily be seen as “dirty” spots on the glass slowed down the speed or temporarily changed the direction of motion of the spheres. The velocity data were not used in such cases.
As seen in Fig.6, in all three cases, the data measured for tilt anglea¼22.5fall clearly below the linear approxi- mation for theUx-values, and the deviations are the largest for the pure gravity andXy<0 data (green and red symbols).
In fact, for this tilt angle, the velocityUdof downhill propul- sion exceeds the velocity due to pure rotational rolling Ux¼aXy–2.6lm/s for a sphere rotating atXy–1.1 s1, and the motion must be different from rolling/slipping and probably not described by Eq.(11).
The measured velocities for the horizontal sample (a¼0) can be compared to those reported by Martinez- Pedreroet al.8,9who reported speeds of 0.6lm/s and 1lm/s for d¼2.8lm spheres at field rotation frequencies of 10 Hz and 150 Hz, respectively, using slightly higher field strength.
These authors developed a model for the translational veloc- ity of single particles and chains of particles purely based on the hydrodynamic interactions among the microspheres.9 Based on their model and the experimental observations, they estimated a glass surface to sphere surface distance of d 180 nm, which is considerably larger than the value found in the present study. In the model used in Ref.9, the density difference between the microspheres and water was neglected, and it was assumed that double layer interactions keep surfaces separated. However, the density of the current microspheres was sufficiently large so that the spheres come in close contact to the glass surface. Since the lift forces are negligible for the low microsphere rotational velocities found here, a rolling/slipping model is the only pattern of motion that can explain the results found in the present experiments.
The value of the friction coefficientlk0.5 may seem higher than what one might expect since, e.g., the friction coefficient for dry polystyrene films sliding against steel has been reported to be 0.6<lk<0.7 (Ref.30) and wetting a surface often reduces sliding friction considerably. Schiweket al.31 measured friction forces of nano-sized polystyrene spheres in water on a silicon wafer using AFM and reported that the adhesion was the strongest for pH¼6 and considerably lower for both higher and lower pH-values. Thus, properties of the substrate itself as well as the ionic strength of solution will influence adhesion and friction. It may be noted that the microscope slides were used without a specific cleaning pro- cedure and a tendency of particles sticking to the surface increased with time.
The observed microsphere motion inside tilted samples opens up possibilities for using such setup for sorting of magnetic microspheres according to particle diameter or sus- ceptibility, as particles with the largest magnetic moment will be able to travel the longest distance when the tilt angle of the sorting cell is slowly increased. Similar types of microbeads are commercially available as streptavidin coated particles that via the streptavidin-biotin reaction can be useful for cell separation and protein or nucleic acid puri- fication.32 Using mixtures of beads with different surface
coatings, which can be attached to distinct cells or macro- molecules, their mobility may depend on the attached cargo, and the differences in ability of the beads to move uphill can then be used for cell/molecule sorting.
V. CONCLUSION
Magnetic propulsion of paramagnetic microspheres near liquid-glass interfaces has been demonstrated. By applying magnetic fields rotating in a plane perpendicular to the inter- face, the magnetic particles in the liquid can be moved in a controlled way in a chosen direction. Using sample cells that are tilted relative to the horizontal direction, microspheres will move uphill, downhill, or be at fixed positions depend- ing on the magnetic parameters (field strength and fre- quency), particle properties (diameter and susceptibility), the sample cell inclination, and the particle-surface contact fric- tion coefficient. Propulsion speeds of 0.5–5lm/s inside hori- zontal cells were found, and these speeds, as well as their tilt angle dependence, are in good agreement with what can be calculated from a rolling-slipping model of the sphere- surface contact with a friction coefficient lk0.50, a slip- ping coefficient of about 0.6, and an effective distance between the spheres’ surface and the glass surface of about 20 nm. The contact friction was partly due to protuberances on the microsphere surface. Such inclined cells containing surface-coated magnetic beads have potential use in biomed- ical separation, e.g., utilizing the streptavidin-biotin reaction to attach a molecular cargo.
SUPPLEMENTARY MATERIAL
See supplementary material for videos showing the motion of two microspheres inside a horizontal sample cell and one microsphere inside a tilted cell.
ACKNOWLEDGMENTS
The Computer Integrated Systems for Microscopy and Manipulation (CISMM) Center at University of North Carolina at Chapel Hill, supported by the NIH NIBIB, is gratefully acknowledged for making the Video Spot Tracker software freely available. A. T. Skjeltorp of IFE/Giamag (Norway) is acknowledged for many stimulating discussions and for useful comments to the manuscript.
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